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Intersection theory in homogeneous spaces of simple Lie groups. (Chinese. English summary) Zbl 1025.22009

Summary: Consider \(H^*(U_n/T_n,\mathbb{Z})\), the integral cohomology ring of the homogeneous space \(U_n/T_n\). It is a quotient ring of the polynomial ring in \(n\) variables. Particularly, the homogeneous part with degree \(d=\dim U_n/T_n\) corresponds to the highest dimensional cohomology group \(\mathbb{Z}\), i.e., each degree \(d\) homogeneous polynomial \(f\) corresponds to an integer \(\chi(f)\). The present article computes this correspondence explicitly. The connections with the intersection theory of the manifold \(U_n/T_n\) are also discussed. A similar result applies to the homogeneous space \(Sp_n/T_n\).

MSC:

22E46 Semisimple Lie groups and their representations
57T15 Homology and cohomology of homogeneous spaces of Lie groups
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